# Should I apply my treatment as a continuous gradient, or in blocks?

I am designing an experiment, and am looking for advice on how to apply my treatment most efficiently.

I'll give an example to explain. My response is car efficiency - how many miles per gallon of fuel the car uses. I have two groups of cars - one from Factory A with a regular factory manager, and one from Factory X, which has a fuel efficiency expert as its manager. There are 20 cars from each factory.

Each factory also has a mixture of ingredients to produce different fuel with, and have produced slightly different fuels for each car. None of the cars have the exact same fuel, but I know the exact ingredients and amounts that have been used.

Finally, I will be testing the cars efficiencies while driving at different speeds, and here is where my main question lies. Should I use a continuous gradient of speeds, or should I use two blocks of speeds?

I have a few hypotheses I want to test:

• Cars from Factory X are more efficient than cars from Factory A.
• The mix of Fuel influences the efficiency of the car.
• Efficiency increases with speed, but more so for cars from Factory X than cars from Factory A.

To test the effect of speed on efficiency I could drive each pair of cars (one from A, one from X) at a different speed, spread evenly between 20mph and 80mph. Alternatively, I could drive half of the cars at 20mph, and half at 80mph. Bear in mind that I have no pre-experiment knowledge on how efficiency varies with speed, but I suspect it is not a linear relationship, probably more of a threshold change.

Which approach would be more powerful for testing my hypotheses?

Another question - I was hoping that a simple regression model would be able to answer most of my questions - something like this:

efficiency ~ Speed + Fuel + Factory


"Fuel" would potentially be a PCA of the fuel ingredients. Would any of the predictors need to be random, or nested effects, or interactions?

Many thanks

• Is there some other factors you want to control? Like drivers? What about a paired experiment, you select pairs of cars, one from each fabric, and drive them at the same time, same conditions, same speed, once, then repeat, switch drivers, the same again? Then you could avoid the need of some specific regression model ... Oct 22, 2020 at 11:27
• Thanks. I've mentioned all of the relevant factors - "driver" is not one of them. I can pair cars from the two factories to have the same speed, but I can't control the "fuel" so that will always differ between the two cars.
– rw2
Oct 22, 2020 at 11:51

## 1 Answer

This depends whether your goal is only to test the hypothesis you tested, or if you are also interested in estimating and criticizing your regression model. First a simple example: If you are interested in a simple regression model $$y=\beta_0 + \beta_1 x+\epsilon$$ with possible values for $$x$$ being the interval $$[a, b]$$, then the optimal design (at least for minimizing prediction variance, but probably also for other criteria), is to take half of observations at $$x=a$$ the other half at $$x=b$$. But, that assumes you trust your model blindly, if the straight line model is wrong, maybe there is some curvature or other nonlinearities, you will not detect that from such a design! So it might be good to allow for a few interior points to allow for model criticism.

This in part answers your question:

Should I use a continuous gradient of speeds, or should I use two blocks of speeds?

With many different values for speed, you allow for nonlinearities and model criticism. You could use Optimal Design theory (in R, package AlgDesign, for an example see Some questions about AlgDesign for Fractional Factorial Design in R) to find good speed values, assuming some flexible model.

But, combine this with a paired design, run the experiment with pairs of matched cars from the two factories. That will also allow for test not depending on the regression model.